Certainly a month of change, as summer makes way for the south-west monsoon, bringing cooler temperatures and more rainfall. In reality June is often a very pleasant month to travel in Myanmar, as the weather can remain relatively dry and the sky-high temperatures of the past few months are subdued. Showers tend to be relatively short lived, increasing in intensity leading up to July.
The central plains to the south of Mandalay remain the hottest and driest region of the country in June, with temperatures still in the low to mid 30s°C. Elsewhere the mercury tends to hover around 30°C, so still far from chilly!
Rain is not unexpected in most corners of Myanmar at this time of year, however you would be unlucky to see much, although the chances are higher at the very end of the month. Most of the beach resorts will be closed by mid June, as they prepare for the less favourable weather conditions expected in the coming months.